Advertisements
Advertisements
Question
Write the domain and range (principle value branch) of the following functions:
f(x) = tan–1 x.
Advertisements
Solution
f(x) = tan–1 x
Domain (– ∞, ∞)
Range `((-π)/2, π/2)`
APPEARS IN
RELATED QUESTIONS
Show that the function f : R → {x ∈ R : –1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.
Give examples of two functions f: N → Z and g: Z → Z such that g o f is injective but gis not injective.
(Hint: Consider f(x) = x and g(x) =|x|)
Which of the following functions from A to B are one-one and onto?
f1 = {(1, 3), (2, 5), (3, 7)} ; A = {1, 2, 3}, B = {3, 5, 7}
Which of the following functions from A to B are one-one and onto?
f2 = {(2, a), (3, b), (4, c)} ; A = {2, 3, 4}, B = {a, b, c}
Classify the following function as injection, surjection or bijection :
f : R → R, defined by f(x) = x3 − x
Give examples of two surjective functions f1 and f2 from Z to Z such that f1 + f2 is not surjective.
Show that f : R→ R, given by f(x) = x — [x], is neither one-one nor onto.
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x and g(x) = |x| .
Find gof and fog when f : R → R and g : R → R is defined by f(x) = x2 + 2x − 3 and g(x) = 3x − 4 .
Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.
Find fog and gof if : f (x) = ex g(x) = loge x .
Let f be a real function given by f (x)=`sqrt (x-2)`
Find each of the following:
(i) fof
(ii) fofof
(iii) (fofof) (38)
(iv) f2
Also, show that fof ≠ `f^2` .
State with reason whether the following functions have inverse :
g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}
Find f −1 if it exists : f : A → B, where A = {1, 3, 5, 7, 9}; B = {0, 1, 9, 25, 49, 81} and f(x) = x2
Write the domain of the real function
`f (x) = 1/(sqrt([x] - x)`.
If f : R → R be defined by f(x) = (3 − x3)1/3, then find fof (x).
\[f : A \to \text{B given by } 3^{ f\left( x \right)} + 2^{- x} = 4\] is a bijection, then
Let
\[A = \left\{ x : - 1 \leq x \leq 1 \right\} \text{and} f : A \to \text{A such that f}\left( x \right) = x|x|\]
The function
\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]
(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto
Let \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]
Let
\[f : [2, \infty ) \to X\] be defined by
\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =
Let [x] denote the greatest integer less than or equal to x. If \[f\left( x \right) = \sin^{- 1} x, g\left( x \right) = \left[ x^2 \right]\text{ and } h\left( x \right) = 2x, \frac{1}{2} \leq x \leq \frac{1}{\sqrt{2}}\]
Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______
Let C be the set of complex numbers. Prove that the mapping f: C → R given by f(z) = |z|, ∀ z ∈ C, is neither one-one nor onto.
The function f : R → R defined by f(x) = 3 – 4x is ____________.
The number of bijective functions from set A to itself when A contains 106 elements is ____________.
Let R be a relation on the set L of lines defined by l1 R l2 if l1 is perpendicular to l2, then relation R is ____________.
Prove that the function f is surjective, where f: N → N such that `f(n) = {{:((n + 1)/2",", if "n is odd"),(n/2",", if "n is even"):}` Is the function injective? Justify your answer.
Let a and b are two positive integers such that b ≠ 1. Let g(a, b) = Number of lattice points inside the quadrilateral formed by lines x = 0, y = 0, x = b and y = a. f(a, b) = `[a/b] + [(2a)/b] + ... + [((b - 1)a)/b]`, then the value of `[(g(101, 37))/(f(101, 37))]` is ______.
(Note P(x, y) is lattice point if x, y ∈ I)
(where [.] denotes greatest integer function)
ASSERTION (A): The relation f : {1, 2, 3, 4} `rightarrow` {x, y, z, p} defined by f = {(1, x), (2, y), (3, z)} is a bijective function.
REASON (R): The function f : {1, 2, 3} `rightarrow` {x, y, z, p} such that f = {(1, x), (2, y), (3, z)} is one-one.
